Ok, so I know that we cannot integrate this with elementary functions as it stands at the moment. Experience with these double integral questions tells me that I should change the order of integration.

Here's my first problem though, and that's figuring out exactly what the new bounds should be (I think I'm getting caught up in all the variables). I sketched the region on an x-t plane, which essentially consisted of a line passing through the origin x=t, and making the observation that x = alpha = t.

Just considering integrating the "cos" function first, here's what I came up with: